Expectation of product of three jointly Gaussian random variables

Question

given $X_1,X_2,X_3∼N(0,σ^2)$ , jointly gaussian with $E[X_iX_j] \ne 0$

Why does $E[X_1X_2X_3]=0 $?

Also, given $X= [X_1,X_2,...X_k]$ , $X_i∼N(0,σ^2)$ jointly gaussian:

Is it true that $E$[ product of n=odd different variables ]$=0$?

Answer 1

I suppose that there should be a simpler proof, but if we can use cumulants, we have that \begin{multline*} \operatorname E[X_1X_2X_3] =\operatorname{cum}[X_1,X_2,X_3]+\operatorname{cum}[X_1]\operatorname{cum}[X_2,X_3]+\operatorname{cum}[X_2]\operatorname{cum}[X_1,X_3]\\ +\operatorname{cum}[X_3]\operatorname{cum}[X_1,X_2] +\operatorname{cum}[X_1]\operatorname{cum}[X_2]\operatorname{cum}[X_3]. \end{multline*} You can find this equality on page 34 of Stationary sequences and random fields by Murray Rosenblatt. Now since all joint cumulants of order greater than $2$ of a Gaussian distribution are $0$, $\operatorname{cum}[X_1,X_2,X_3]=0$. Also, $\operatorname{cum}[X_i]=\operatorname EX_i=0$ for $i=1,2,3$. Hence, $\operatorname E[X_1X_2X_3]=0$.

Using similar arguments, we can show that a product of odd number of random variables with joint Gaussian distribution is also equal to $0$.